Difficulty: Intermediate | Prerequisites: Limits and Continuity (Sections 2.1–2.3), algebra rules for exponents and logarithms.
Derivatives answer the question "how fast is this changing right now?" They grow directly out of the limit concept: the derivative is the limit of the average rate of change as the interval shrinks to zero. Once you know how to take derivatives, you can find slopes of tangent lines, determine where functions increase or decrease, and solve business problems about cost, revenue and profit. This block of material covers the definition, the shortcut rules (power, product, quotient, chain), and applications to marginal analysis and exponential models.
The derivative f′(x) gives the instantaneous rate of change, which is the slope of the tangent line. You find it either from the limit definition or by applying rules (power, product, quotient, chain). In business contexts, the derivative of a total cost, revenue or profit function gives the marginal function, which estimates the cost, revenue or profit of the next unit.
Secant line
A line connecting two points on a curve. Its slope equals the average rate of change between those points.
Tangent line
A line that touches the curve at exactly one point and has the same slope as the curve at that point. Its slope equals the instantaneous rate of change.
Average rate of change
The slope of the secant line: [f(x₂) − f(x₁)] / (x₂ − x₁). In simple terms: how much the output changed per unit of input over an interval.
Instantaneous rate of change
The limit of the average rate of change as the interval shrinks to zero. Equals the derivative at that point.
Derivative
f′(x) = lim h→0 [f(x + h) − f(x)] / h. It gives the slope of the tangent line and the instantaneous rate of change at any x.
In simple terms: the speedometer reading, not the trip average.
Marginal cost / revenue / profit
The derivative of the total cost, revenue or profit function. C′(x) estimates the cost of producing one more unit, R′(x) estimates the revenue from selling one more unit, and P′(x) estimates the profit on one more unit.
Compound interest
A = P(1 + r/m)^(mt), where P is principal, r is annual rate, m is compounding periods per year, and t is time in years.
Continuous compounding
A = Pe^(rt). The limiting case as compounding frequency goes to infinity. The number e (≈ 2.71828) arises naturally from lim n→∞ (1 + 1/n)^n.
Average rate of change (slope of the secant line through two points):
m_sec = [f(x₂) − f(x₁)] / (x₂ − x₁) = [f(a + h) − f(a)] / h
where h = x₂ − x₁.
Instantaneous rate of change (slope of the tangent line at one point):
m_tan = lim h→0 [f(a + h) − f(a)] / h
This is what the derivative computes.
Definition of the derivative (the four-step process):
f′(x) = lim h→0 [f(x + h) − f(x)] / h
Step 1: Find f(x + h) by replacing every x in f with (x + h). Step 2: Compute f(x + h) − f(x) and simplify. Step 3: Divide by h and cancel h. Step 4: Take the limit as h → 0.
Worked example: f(x) = x² − x + 2.
Step 1: f(x + h) = (x + h)² − (x + h) + 2 = x² + 2xh + h² − x − h + 2.
Step 2: f(x + h) − f(x) = 2xh + h² − h.
Step 3: (2xh + h² − h)/h = 2x + h − 1.
Step 4: lim h→0 (2x + h − 1) = 2x − 1.
So f′(x) = 2x − 1.
Key identity: DERIVATIVE = INSTANTANEOUS RATE OF CHANGE = SLOPE OF THE TANGENT LINE. These three phrases all mean the same thing.
Constant Rule: d/dx [c] = 0. The derivative of any constant is zero.
Line Rule: d/dx [cx] = c. The derivative of cx is just the coefficient.
Power Rule: d/dx [x^n] = nx^(n−1). Bring the exponent down as a coefficient, then reduce the exponent by one.
Before differentiating, rewrite roots and fractions as powers:
1/x^n = x^(−n)
ⁿx√(x^m) = x^(m/n)
Do not write f′(x) or y′ until you start the actual derivative. The algebra rewrite is separate.
Examples:
f(x) = √x = x^(1/2). Then f′(x) = (1/2)x^(−1/2) = 1/(2√x).
y = 1/x³ = x^(−3). Then y′ = −3x^(−4) = −3/x⁴.
f(x) = (2x − 3)(x² − 4x). Expand first: 2x³ − 8x² − 3x² + 12x = 2x³ − 11x² + 12x. Then f′(x) = 6x² − 22x + 12.
y = (7x − 6x⁵ + 2)/x². Split: 7x^(−1) − 6x³ + 2x^(−2). Then y′ = −7x^(−2) − 18x² − 4x^(−3).
Finding tangent line equations:
To find the equation of the tangent line to f(x) at x = c:
Compute f(c) to get the y-coordinate (the point is (c, f(c))).
Compute f′(c) to get the slope m.
Use point-slope form: y − f(c) = f′(c)(x − c).
This problem type appears on every exam.
Finding horizontal tangent lines:
A horizontal tangent has slope zero, so set f′(x) = 0 and solve for x.
Profit = Revenue − Cost: P(x) = R(x) − C(x).
Revenue = (quantity)(price): R(x) = x · p(x).
Marginal functions are the derivatives of the total functions:
Marginal cost: C′(x).
Marginal revenue: R′(x).
Marginal profit: P′(x).
Exact cost of the nth unit: C(n) − C(n − 1). This is the actual difference.
Estimated cost of the nth unit: C′(n − 1). Evaluate the marginal cost at one unit less. This gives an approximation.
Average functions divide the total by x:
Average cost: C̄(x) = C(x)/x.
Average revenue: R̄(x) = R(x)/x.
Average profit: P̄(x) = P(x)/x.
The marginal average functions are the derivatives of the average functions: C̄′(x), R̄′(x), P̄′(x). Find the average function first, then differentiate.
Always label your functions with proper notation (C′(x) =, R′(x) =, etc.).
The number e:
lim n→∞ (1 + 1/n)^n = e ≈ 2.71828.
Compound interest: A = P(1 + r/m)^(mt).
m = 1 (annually), m = 2 (semi-annually), m = 12 (monthly), m = 365 (daily).
Continuous compounding: A = Pe^(rt).
Continuous decay: Same formula with a negative rate r.
Doubling/tripling time: Set A = 2P (or 3P), divide both sides by P to get 2 = e^(rt), take ln of both sides: t = ln(2)/r (or t = ln(3)/r).
Derivative rules for e and ln:
d/dx [e^x] = e^x (the function is its own derivative).
d/dx [e^(Cx)] = Ce^(Cx) where C is a constant.
d/dx [ln x] = 1/x.
Algebra rules you need before differentiating:
ln(ab) = ln a + ln b.
ln(a/b) = ln a − ln b.
ln(a^p) = p ln a.
ln(e^x) = x.
Always simplify using these log rules before taking the derivative.
Examples:
f(x) = ln(x⁴) + e^(3x) − 45. Simplify: 4 ln x + e^(3x) − 45. Derivative: f′(x) = 4/x + 3e^(3x).
f(x) = 4 ln(5x) + 2x^e − e^(−5x). Simplify: 4(ln 5 + ln x) + 2x^e − e^(−5x) = 4 ln 5 + 4 ln x + 2x^e − e^(−5x). Derivative: f′(x) = 4/x + 2ex^(e−1) + 5e^(−5x). Note: 2x^e uses the power rule because e is just a constant exponent.
Product Rule: (F · S)′ = F′S + FS′. First derivative times second, plus first times second derivative.
Label F, S and their derivatives on the side before plugging in.
Example: f(x) = (x² − 4x + 1)(x² − 1). F = x² − 4x + 1, F′ = 2x − 4, S = x² − 1, S′ = 2x. Then f′(x) = (2x − 4)(x² − 1) + (x² − 4x + 1)(2x).
Quotient Rule: (T/B)′ = (T′B − TB′) / B². Top derivative times bottom, minus top times bottom derivative, all over bottom squared.
Label T, B and their derivatives on the side before plugging in.
Example: f(x) = x/(4x − 8). T = x, T′ = 1, B = 4x − 8, B′ = 4. Then f′(x) = [(1)(4x − 8) − (x)(4)] / (4x − 8)² = (4x − 8 − 4x) / (4x − 8)² = −8/(4x − 8)².
When to use which:
Two expressions multiplied together, neither of which simplifies by expanding easily: product rule.
A fraction where you cannot split terms: quotient rule.
If you can expand or split the fraction into simpler terms first, do that and use the power rule instead.
Finding horizontal tangents with these rules: Differentiate using the appropriate rule, set the numerator of f′(x) equal to zero (since a fraction is zero when its numerator is zero), and solve.
Chain Rule: If y = f(g(x)), then y′ = f′(g(x)) · g′(x). Differentiate the outside function, leave the inside alone, then multiply by the derivative of the inside.
General Power Rule (special case of chain rule):
d/dx [u^n] = nu^(n−1) · u′, where u = u(x) is any function of x.
Chain rule with e and ln:
d/dx [e^u] = e^u · u′.
d/dx [ln u] = u′/u.
How to apply it:
Identify the inner function u(x) and compute u′(x).
Differentiate the outer function as if u were just x.
Multiply the result by u′(x).
Examples:
f(x) = (4x + 1)⁵. Inner: u = 4x + 1, u′ = 4. f′(x) = 5(4x + 1)⁴ · 4 = 20(4x + 1)⁴.
y = e^(3x² − 1). Inner: u = 3x² − 1, u′ = 6x. y′ = e^(3x² − 1) · 6x = 6xe^(3x² − 1).
y = ln(4x² + 5x + 3). Inner: u = 4x² + 5x + 3, u′ = 8x + 5. y′ = (8x + 5)/(4x² + 5x + 3).
y = (x + 3)²(2x − 5)³. Product rule combined with chain rule: y′ = 2(x + 3)(2x − 5)³ + (x + 3)² · 3(2x − 5)² · 2.
Logarithmic differentiation is useful when the function has products, quotients, and powers nested together.
Steps:
Take ln of both sides: ln y = ln f(x).
Use log rules to expand: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln(a^r) = r ln a.
Differentiate both sides. The left side gives y′/y.
Multiply both sides by y (which is the original f(x)).
Example: f(x) = e^(4x)(3x + 1)² / √(5x² − 2).
ln y = 4x + 2 ln(3x + 1) − (1/2) ln(5x² − 2).
y′/y = 4 + 6/(3x + 1) − 5x/(5x² − 2).
y′ = f(x) · [4 + 6/(3x + 1) − 5x/(5x² − 2)].
Differentiation rules:
Constant: [c]′ = 0
Line: [cx]′ = c
Power: [x^n]′ = nx^(n−1)
e: [e^x]′ = e^x
e^(Cx): [e^(Cx)]′ = Ce^(Cx)
ln: [ln x]′ = 1/x
Product: [FS]′ = F′S + FS′
Quotient: [T/B]′ = (T′B − TB′)/B²
Chain (power): [u^n]′ = nu^(n−1) · u′
Chain (e): [e^u]′ = e^u · u′
Chain (ln): [ln u]′ = u′/u
Rate of change:
Average: [f(b) − f(a)] / (b − a)
Instantaneous: f′(a) = lim h→0 [f(a + h) − f(a)] / h
Tangent line: y − f(c) = f′(c)(x − c)
Marginal analysis:
Exact nth unit: C(n) − C(n − 1)
Estimated nth unit: C′(n − 1)
Interest:
Compound: A = P(1 + r/m)^(mt)
Continuous: A = Pe^(rt)
Doubling time: t = ln 2 / r
Marginal analysis is central to business decision-making: a company uses C′(x) to decide whether producing one more unit is worth the cost. Derivatives of continuous-compounding models (A′ = rPe^(rt)) give the instantaneous rate at which an investment is growing, used in finance to evaluate returns.
Students confuse the average rate of change with the instantaneous rate. The average uses two points; the derivative uses one.
Students forget to rewrite roots and fractions as powers before differentiating. You cannot apply the power rule to 1/x³ until you write it as x^(−3).
Students apply the product or quotient rule when expansion or splitting would be simpler and less error-prone.
Students forget the chain rule. If there is a function inside another function, the chain rule applies. The most common error is omitting the u′ factor.
⚠️ "Find the equation of the tangent line of f(x) at x = c" appears on every exam. Know the three steps cold.
⚠️ Marginal analysis (exact vs estimated) is tested regularly.
⚠️ Always simplify with log rules before differentiating ln expressions.
⚠️ Label all functions: write C′(x) =, not just the expression.
True or false: The derivative of e^(3x) is e^(3x).
Fill in the blank: The derivative of ln(x⁴) after simplification is ________.
True or false: C′(10) estimates the cost of the 10th unit.
Fill in the blank: To find the tangent line at x = c, you need f(c) for the ________ and f′(c) for the ________.
True or false: d/dx [√x] = 1/(2√x).
Answers: 1. False (it is 3e^(3x)). 2. 4/x. 3. False (it estimates the 11th unit; use C′(n − 1) for the nth). 4. y-coordinate, slope. 5. True.
Q: Find f′(x) for f(x) = 2x² ln(x).
A: Product rule. F = 2x², F′ = 4x, S = ln x, S′ = 1/x. f′(x) = 4x ln x + 2x²(1/x) = 4x ln x + 2x.
Q: A company’s total cost is C(x) = x² + 20x + 5000. What is the exact cost of the 11th unit?
A: C(11) − C(10) = (121 + 220 + 5000) − (100 + 200 + 5000) = 5341 − 5300 = $41.
Q: Use marginal analysis to estimate the cost of the 11th unit.
A: C′(x) = 2x + 20. C′(10) = 40. Estimated cost is $40.
Q: Find the tangent line to f(x) = x² − x + 2 at x = 1.
A: f(1) = 1 − 1 + 2 = 2 (point is (1, 2)). f′(x) = 2x − 1, so f′(1) = 1 (slope). y − 2 = 1(x − 1), so y = x + 1.
Q: If $200 is invested at 3% compounded continuously, how long to double?
A: t = ln 2 / 0.03 ≈ 23.1 years.
Derivatives are the foundation for curve sketching (Sections 4.1, 4.2, 4.4), where the first derivative reveals increasing/decreasing intervals and the second derivative reveals concavity. Optimisation (Sections 4.5, 4.6) uses derivatives to find maximum profit or minimum cost. Integration (Sections 5.1, 5.5) reverses the derivative.
derivative, differentiation, power rule, product rule, quotient rule, chain rule, general power rule, tangent line, secant line, slope, average rate of change, instantaneous rate of change, marginal cost, marginal revenue, marginal profit, average cost, compound interest, continuous compounding, exponential growth, exponential decay, doubling time, e, ln, natural log, logarithmic differentiation, four-step process, MAC 2233, business calculus, FIU, Sections 2.4, 2.5, 2.7, 3.1, 3.2, 3.3, 3.4